Rogue peakon, posedness and blow-up phenomenon for an integrable Camassa–Holm type equation

Authors

Mingxuan Zhu
Zhenteng Zeng
Zaihong Jiang
Baoqiang Xia
Zhijun Qiao, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

1-20-2026

Abstract

In this paper, we study an integrable Camassa–Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution – called rogue peakon, that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space 𝐵𝑠𝑝,𝑟 with 1 ≤ p, r ≤ ∞, 𝑠 >max⁡{1+1/𝑝,3/2} or 𝐵3/22,1 , and then prove the ill-posedness of the solution in 𝐵3/22,∞ . Moreover, we establish the global existence and blow-up phenomenon of the solution, which is, if m 0(x) = u 0 − u 0xx ≥ (≢)0, then the corresponding solution exists globally, meanwhile, if m 0(x) ≤ (≢)0, then the corresponding solution blows up in a finite time.

Comments

© 2026 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

Recommended Citation

Zhu, Mingxuan, Zhijun Qiao, Zhidong Wang, and Zaihong Jiang. 2026. “Rogue Peakon, Posedness and Blow-up Phenomenon for an Integrable Camassa–Holm Type Equation.” Advances in Nonlinear Analysis 15 (1). https://doi.org/10.1515/anona-2025-0139.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication Title

Advances in Nonlinear Analysis

DOI

10.1515/anona-2025-0139